«CooPeR»
BIPS — LMU/SLDS — MCML
2023-05-12
Main goal: Fit cause-specific model for event 1 using shared information from event 2
The elastic net objective function with some negative log-likelihood term:
\underset{\beta}{\operatorname{arg min}} \quad \mathrm{NLL}(\beta) + {\color{blue}\lambda} \sum_{j=1}^p \left( {\color{red}\alpha} |\beta_j| + \frac{1 - {\color{red}\alpha}}{2} \beta^2_j \right)
fwelnet)Example for p = 5 features X_{1,2,3,4,5} and K = 2 groups
\mathbf{Z} = \begin{pmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \\ 0 & 1 \\ \end{pmatrix}
Example for p = 4 features X_{1,2,3,4}:
\mathbf{Z} = \begin{pmatrix} 0.5 \\ 1 \\ 2 \\ 0 \\ \end{pmatrix}
\underset{\beta}{\operatorname{arg min}} \quad \mathrm{NLL}(\beta) + \lambda \sum_{j=1}^p {\color{blue}w_j(\theta)}\left( \alpha |\beta_j| + \frac{1 - \alpha}{2} \beta^2_j \right)
{\color{blue}w_j(\theta)} = \frac{\sum_{l=1}^p \exp(\mathbf{z}_l^T \theta)}{p \exp(\mathbf{z}_j^T \theta)}
Set \beta_1^{(0)}, \beta_2^{(0)} to glmnet solution for (\mathbf{X}, \mathbf{y}_1), (\mathbf{X}, \mathbf{y}_2) respectively
For k = 0, 1, \ldots:
fwelnet with (\mathbf{X}, \mathbf{y}_2, \mathbf{Z}_2)
lambdafwelnet with (\mathbf{X}, \mathbf{y}_1, \mathbf{Z}_1)
lambdaDubbed “Cooperative Penalized (Cox) Regression” (CooPeR)
Block 1 (Mutual): \rho \approx 0.5
same effect (0.5) on both causes
Block 2 (Reversed): \rho \approx 0.35
positive effect on cause 1 and negative on cause 2
Block 3 (Disjoint): \rho \approx 0.05
3.1: effect on cause 1 only
3.2: effect on cause 2 only
Block 4 (Cor. Noise): 500 variables, \rho \approx 0.32
Remaining variables: Uncorrelated noise
\mathrm{PPV} = \frac{\mathrm{TP}}{\mathrm{TP} + \mathrm{FP}}
\mathrm{PPV} = \frac{\mathrm{TP}}{\mathrm{TP} + \mathrm{FP}}
\mathrm{FPR} = \frac{\mathrm{FP}}{\mathrm{FP} + \mathrm{TN}} = 1 - \mathrm{TNR}
\mathrm{FPR} = \frac{\mathrm{FP}}{\mathrm{FP} + \mathrm{TN}} = 1 - \mathrm{TNR}
Thanks for listening!